ECON 3818

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# ECON 3818
## Chapter 15
### Kyle Butts
### 26 July 2021

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## Chapter 15: Parameters and Statistics

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# Parameters and Statistics
We have discussed using sample data to make inference about the population.  In particular, we will use sample .hi.green[statistics] to make inference about population .hi.purple[parameters].

A .hi.purple[parameter] is a number that describes the population. In practice, parameters are unknown because we cannot examine the entire population.

A .hi.green[statistic] is a number that can be calculated from sample data without using any unknown parameters. In practice, we use statistics to estimate parameters.

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# Greek Letters and Statistics

.pull-left[
.hi.green[Latin Letters]

- Latin letters like `$\bar{x}$` and `$s^2$` are calculations that represent guesses (estimates) at the population values.
]
.pull-right[
.hi.purple[Greek Letters]

- Greek letters like `$\mu$` and `$\sigma^2$` represent the truth about the population.
]

The goal for the class is for the latin letters to be good guesses for the greek letters:

$$
	{\color{green}\text{Data}} \longrightarrow {\color{green}\text{Calculation}} \longrightarrow {\color{green}\text{Estimates}} \longrightarrow^{hopefully!} {\color{purple}\text{Truth}}
$$

For example, 
$$
	{\color{green}X} \longrightarrow {\color{green} \frac{1}{n} \sum_{i=1}^n X_i} \longrightarrow {\color{green}\bar{x}} \longrightarrow^{hopefullly!} {\color{purple}\mu}
$$

---
# Examples of Parameters
Some parameters of distributions we've encountered are

- `$n$` and `$p$` in `$X\sim B(n,p)$` with probability mass function
$$
	P(X=x)={n \choose x} p^x \left(1-p\right)^{n-x}
$$

- `$a$` and `$b$` in `$X\sim U(a,b)$` with probability density function
$$
	f(x)=\frac{1}{b-a}
$$

- `$\mu$` and `$\sigma^2$` in `$X\sim N(\mu,\sigma^2)$` with probability density function

$$
	f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\left(\frac{x-\mu}{\sigma}\right)^2} 
$$

---
# Mean and Variance

Two population parameters of particular interest are

- the mean, denoted `$\mu$`, defined by `$E(X)$`
- the variance, denoted `$\sigma^2$`, defined by `$E(X^2)-E(X)^2$`

We .hi[do not] observe these. Therefore, we guess using

- the sample mean, `$\bar{X}$`
- the sample variance, `$s^2$`

Why do we use these as our guess?

---
# Getting the right sample
Before we talk about the properties of sample statistics, we need to make sure we have the right sample. We talked about good ways to generate a sample.

.hi.it[The right sample is the most important part of any data analysis.]

A .hi.green[Simple Random Sample] has no bias and has observations that are from the same population.

---
# Identically Distributed

If every observation is from the same population, we say all of the observations in our sample are .hi.red_pink[identically distributed].  In math, this means for any two observations `$X_i$` and `$X_j$`,

$$
	Pr(X_i < x) = Pr(X_j < x)
$$

---
# Independent Observations

Does observing `$X_i$` impact our best guess of `$X_j$`?  Sometimes yes (time series, spatial dependence), but hopefully not.

To simplify things, we need to assume .hi.red[independent sample observations], meaning 
$$
	Pr(X_i=a \ \vert \ X_j=b) = Pr(X_i=a)
$$

Intuitively, this means that .it[observing] one outcome doesn't help you .it[predict] any other outcome.

To summarize, we want an .it[i.i.d.] sample, i.e. sample observations that are .hi.purple[independent and identically distributed].

---
# Sample Statistics are Random Variables

For a sample `$X_1,..., X_n$` of the random variable `$X$`, any function of that sample, `$\hat{\theta}=g(X_1,...,X_n)$`, is a .hi.turquoise[sample statistic].  For example,

`$${\color{turquoise}\bar{X}} = \frac{1}{n} \sum_{i=1}^{n} X_i$$`

`$${\color{turquoise}\displaystyle s^2} = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2$$`

Because `$X_1,..., X_n$` are random variables, any sample statistic `${\color{turquoise} \hat{\theta}} = g(X_1,...,X_n)$` is itself a random variable!

That means, there is some distribution for the values of `${\color{turquoise} \hat{\theta}}$`

---
# Sampling Distributions

This is one of the most important concepts in the course. One .hi[trial] would consist of the following:

- .hi.green[Random Sample] - Grab a group of observations from the population

- .hi.turquoise[Sample Statistic] - Take your particular random sample and calculate a sample statistic (e.g. sample mean)

.hi.orange[Sampling Distribution] - Imagine repeatedly grabbing a different group of observations from the population and calculating the sample mean. This is performing many .hi[trials]. The sample means themselves will have a distribution.

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.center[
<img style="width:100%;" src="data:image/png;base64,#sample_dist.gif"/>
]

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---
# Sample Size

The variance of the .it.orange[sampling distribution] depends on the sample size. As `$n$` gets larger, each individual .hi[trial] gives a better guess at the mean. Hence, the .orange[sampling distribution] gets more narrow

.center[
<img style="width:80%;" src="data:image/png;base64,#dist_n.gif"/>
]

---
# Sampling Distributions

---
# Sampling Distributions
We will only observe 1 sample in the world though.

How does the concept of .orange[sampling distribution] help us?

- Since we don't know the true population parameter, Our .turquoise[sample statistic] will be our best guess at the possible true value.

- If we know the .orange[sampling distribution], then we can consider uncertainty about our .turquoise[sample statistic].

---
# Law of Large Numbers

Is `$\bar{X}$` actually a good guess for `$\mu$`? Under certain conditions, we can use the .hi.purple[Law of Large Numbers (LLN)] to guarantee that `$\bar{X}$` approaches `$\mu$` as the sample size grows large.

.hi[Theorem]: Let `$X_1,X_2,...,X_n$` be an i.i.d. set of observations with `$E(X_i) = \mu$`.

Define the sample mean of size `$n$` as `$\bar{X}_n = \frac{1}{n}\sum_{i = 1}^{n}X_i$`. Then

$$ \bar{X}_n \to \mu \quad \text{as} \quad n \to \infty. $$

Intuitively, as we observe a larger and larger sample, we average over randomness and our sample mean approaches the true population mean.

---
# Law of Large Numbers

.center[
<img style="width:80%;" src="data:image/png;base64,#lln.gif"/>
]

---
# Law of Large Numbers

---
# Properties of the sample mean

.hi[Theorem]:
Let `$X_1,X_2,...,X_n$` be an i.i.d. sample with `$E(X_i) = \mu$` and `$Var(X_i) = \sigma^2 < \infty$`. Then

`$$E(\bar{X}_n) = \mu$$`

`$$Var(\bar{X}_n) = \frac{\sigma^2}{n}$$`

Intuitively, we grab many samples from a population. The variance of our sample averages shrinks as we observe more observations per sample.

---
# Clicker Question
Suppose we sample 100 observations from a distribution with `$\mu = 15$` and `$\sigma^2 = 25$`. What are `$E(\bar{X}_{100})$` and `$Var(\bar{X}_{100})$`?

---
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## When is the sample mean Normally Distributed?

Although we know the mean and variance of `$\bar{X}$`, we generally don't know its distribution function.

.hi[Theorem]: Let `$X_1,X_2,...,X_n$` be an i.i.d. sample with `$X_i \sim N(\mu, \sigma^2)$` for `$i=1,2,...,n$`.

Then 
$$
	\bar{X}_n \sim N(\mu, \frac{\sigma^2}{n}).
$$

Intuitively, if all the observations come from the same normal distribution then the sample average is also normally distributed and centered at the true mean (but much more narrow).

---
# Central Limit Theorem

What if `$X_i$` are not normally distributed?

If the number of observation, `$n$`, per sample is large (we will discuss this more later), then the distribution of `$X_i$` doesn't matter. We will always have

$$
	\bar{X}_n \sim N(\mu, \frac{\sigma^2}{n}).
$$